.center[Variation in Naive T cell pool size]

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• Counts of \(\small \text{CD}44^{low} \;\; \text{CD62L}^{high}\) cells isolated from pooled lymph nodes.

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.center[Measuring cell division in T lymphocytes]

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• \(\small \text{Ki}67\) — a nuclear protein expressed during cell-division.

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.pull-left[ ] .pull-right[
- T cells express Ki67 protein for ~ 4 days after the division. .right[Hogan et. al., PNAS 2015]] - Hard to untangle the origin of Ki67 expression within peripheral naive T cells. Thymic Inheritance or division in periphery?

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Mathematical models

• can track the division history of naive T cells and quantify the contributions to Ki67 expression from thymic influx and peripheral division.

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• can dissect the heterogeneity in ‘homeostatic fitness’ amongst naive T cells.

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.center[Potential mechanisms of naive T cell maintenance]

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Model vaildation and comparison

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• Each model was fitted simultaneously to the timecourses of \(\small log(\text{counts})\) proportions of \(\small logit(\text{Ki}67^{high})\) cells.

\[ \text{Joint Likelihood} \Rightarrow p(y | \hat \theta) = \prod_{i=1}^n \, \text{Normal}(y_i - \text{model}(\theta)_i^\text{pred}, \, \sigma) \]

??? We form a joint likelihood for each model assuming the errors were normally distributed.

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• Update prior beliefs \(p(\theta_\text{prior})\) in model parameter using bayesian approach.

\[ p(\hat \theta | y) = \frac{p(y | \hat \theta) \times p(\theta_\text{prior})} {p(y)} \]

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Models are selected based on parsimony and their ability to predict New observations — (out-of-sample prediction error).

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.left[Fits for CD4 data]

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.left[Fits for CD8 data]

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.left[Two-compartment models]

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Adaptation: age-structured models

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Naive T cells of age ‘a’ and Ki67 expression ‘k’ are tracked over time by analytically solving the 3 dimensional PDE,

.shadedbox[ \[ \begin{aligned} \frac{\partial N}{\partial t} + \frac{\partial N}{\partial a} - \beta \, k \, \frac{\partial N}{\partial k}= -(\rho(a) + \delta(a)) \, N(t, a, k) \end{aligned} \]]

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Thanks!

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